As it turns out, for each \(K\)  there is a conjectured
critical threshold \(\alpha _{\rm SAT}=M/N\)  for the
satisfiability property, which was rigorously proven [1]} for large enough \(K\) .
In particular, for every \(\alpha <\alpha _{\rm SAT}\) , the formula admits a satisfying
assignment when \(M/N<\alpha \) , and for every \(\alpha >\alpha _{\rm SAT}\) ,
the formula does not admit a satisfying assignment when \(M/N>\alpha \) , both
with overwhelming probability as \(N\rightarrow \infty \) .
The sharpness of this transition was established rigorously earlier
by general methods in [2]}.
The algorithmic counterparts, however, came short of achieving
the value \(\alpha _{\rm SAT}\)  for every \(K\ge 3\)  (more on this below).
Even before the results in [1]} and [2]}, it was conjectured
in the physics literature that perhaps the existence
of the sharp phase transition property itself
is the culprit for the algorithmic hardness [5]},
[6]}. This was argued by studying the heuristic running times of finding the satisfying
solutions or demonstrating non-existence of ones for small values of \(K\) ,
and observing that the running time explodes near \(\alpha _{\rm SAT}\) , and subsides
when \(M/N\)  is far from \(\alpha _{\rm SAT}\)  on either side. The values of
\(\alpha _{\rm SAT}\)  are known thanks to the powerful machinery of the replica symmetry methods developed in a seminal works of
Parisi [7]},[8]}. These methods give very precise
predictions for the values of \(\alpha _{\rm SAT}\)  for every \(K\) .
For example \(\alpha _{\rm SAT}\)  is approximately
\(4.26\)  when \(K=3\) . An in depth analysis of the exponent of the running times was reported in [5]}.
